Internet Mathematics Articles (Project Euclid)

Local Partitioning for Directed Graphs Using PageRank

Thu, 05 Aug 2010 15:41 EDT

Reid Andersen, Fan Chung, Kevin Lang

Source: Internet Math., Volume 5, Number 1-2, 3--22.

Abstract:
A local partitioning algorithm finds a set with small conductance near a specified seed vertex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set $S$ that has conductance at most $\alpha$, and by performing a sweep over that vector, we can obtain a set of vertices $S'$ with conductance $\Phi_{M}(S')= O(\sqrt{\alpha \log |S|})$. Here, the conductance function $\Phi_{M}$ is defined in terms of the stationary distribution of a random walk in the directed graph. In addition, we describe how this algorithm may be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to partition directed graphs that are not strongly connected.

The Structure of PEC Networks

Wed, 08 Sep 2010 15:15 EDT

Dana Richards, Zhenlei Jia

Source: Internet Math., Volume 6, Number 1, 3--17.

Abstract:
A packed exponential connections (PEC) network is a grid-based network with connectivity and routing results that are competitive with hypercubic networks. The prior results are all empirical, since the structure of the network has been understood only through an indirect existence proof. In this paper we provide the first direct characterization of a PEC network.

An Occupancy Problem Arising in Power Law Fitting

Wed, 08 Sep 2010 15:15 EDT

Ian Abramson, Arthur Berg

Source: Internet Math., Volume 6, Number 1, 19--28.

Abstract:
The power law arises commonly in modeling the number of vertices of a given degree in large graphs. In estimating the degree of the power law, the typical approach is to truncate by eye the log-log plot, then fit a linear equation to the remaining log-transformed data. Here we formulate a hard-coded truncation rule to replace the visual truncation, justify it by showing that the truncation point goes to infinity and misses a vanishing fraction of the data with probability tending to one, and refine the subsequent regression with a weighting and a way to use the covariation between slope and intercept to optimize the slope estimate.

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Wed, 08 Sep 2010 15:15 EDT

Jure Leskovec, Kevin J. Lang, Anirban Dasgupta, Michael W. Mahoney

Source: Internet Math., Volume 6, Number 1, 29--123.

Abstract:
A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as “real” communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best” possible community—according to the conductance measure—over a wide range of size scales. We study over one hundred large real-world networks, ranging from traditional and online social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes. Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually “blend into” the expander-like core of the network and thus become less “community-like,” with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number, which gives a limit to the size of a well-functioning community. However, this behavior is not explained, even at a qualitative level, by any of the commonly used network-generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as test beds of community-detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network data sets.

The Price of Malice: A Game-Theoretic Framework for Malicious Behavior

Fri, 24 Sep 2010 10:38 EDT

Thomas Moscibroda, Stefan Schmid, Roger Wattenhofer

Source: Internet Math., Volume 6, Number 2, 125--156.

Abstract:
In recent years, game theory has provided insights into the behavior of distributed systems by modeling the players as utility-maximizing agents. In particular, it has been shown that selfishness causes many systems to perform in a globally suboptimal fashion. Such systems are said to have a large price of anarchy . In this article, we extend this field of research by allowing some players to be malicious rather than selfish. What, we ask, is the impact of malicious players on the system consisting of otherwise selfish players? In particular, we introduce the price of malice as a measure that captures how much the system's efficiency degrades in the presence of malicious players, compared to a purely selfish environment. As a specific example, we analyze the price of malice of a game that models the containment of the spread of viruses. In this game, each player or node can choose whether to install antivirus software. Then, a virus starts from a random node and recursively infects all neighboring nodes that are not inoculated. We establish various results about this game. For instance, we quantify how much the presence of malicious players can deteriorate or---in case of highly risk-averse selfish players---improve the social welfare of the distributed system.

On Accelerating the PageRank Computation

Fri, 24 Sep 2010 10:38 EDT

Steve Osborne, Jorge Rebaza, Elizabeth Wiggins

Source: Internet Math., Volume 6, Number 2, 157--172.

Abstract:
In this paper we consider the problem of computing the PageRank vector in an efficient way. By combining some of the existing techniques and different approaches, including the power method, linear systems, iterative aggregation/disaggregation, and matrix reorderings, we propose algorithms that decrease the number of iterations to reach the desired solution, thus accelerating convergence to the vector that contains the importance of web pages.

Spectral Properties of the Threshold Network Model

Fri, 24 Sep 2010 10:38 EDT

Yusuke Ide, Norio Konno, Nobuaki Obata

Source: Internet Math., Volume 6, Number 2, 173--188.

Abstract:
We study the spectral distribution of the threshold network model. The results contain an explicit description of the distribution and its asymptotic behavior.

Random Alpha PageRank

Fri, 24 Sep 2010 10:38 EDT

Paul G. Constantine, David F. Gleich

Source: Internet Math., Volume 6, Number 2, 189--236.

Abstract:
We suggest a revision to the PageRank random surfer model that considers the influence of a population of random surfers on the PageRank vector. In the revised model, each member of the population has its own teleportation parameter chosen from a probability distribution, and consequently, the ranking vector is random. We propose three algorithms for computing the statistics of the random ranking vector based respectively on (i) random sampling, (ii) paths along the links of the underlying graph, and (iii) quadrature formulas. We find that the expectation of the random ranking vector produces similar rankings to its deterministic analogue, but the standard deviation gives uncorrelated information (under a Kendall-tau metric) with myriad potential uses. We examine applications of this model to web spam.

Distributing Antidote Using PageRank Vectors

Fri, 24 Sep 2010 10:38 EDT

Fan Chung, Paul Horn, Alexander Tsiatas

Source: Internet Math., Volume 6, Number 2, 237--254.

Abstract:
We give an analysis of a variant of the contact process on finite graphs, allowing for nonuniform cure rates, modeling antidote distribution. We examine an inoculation scheme using PageRank vectors that quantify the correlations among vertices in the contact graph. We show that for a contact graph on $n$ nodes we can select a set $H$ of nodes to inoculate such that with probability at least $1-2\ep$, any infection from any starting infected set of $s$ nodes will die out in $c \log s + c'$ time, where $c$ and $c'$ depend only on the probabilistic error bound $\ep$ and the infection rate, and the size of $H$ depends only on $s$, $\ep$, and the topology around the initially infected nodes, independent of the size of the whole graph.

Permuting Web and Social Graphs

Mon, 10 Oct 2011 13:58 EDT

Paolo Boldi, Massimo Santini, Sebastiano Vigna

Source: Internet Math., Volume 6, Number 3, 257--283.

Abstract:
Since the first investigations on web-graph compression, it has been clear that the ordering of the nodes of a web graph has a fundamental influence on the compression rate (usually expressed as the number of bits per link). The authors of the LINK database, for instance, investigated three different approaches: an extrinsic ordering (URL ordering) and two intrinsic orderings based on the rows of the adjacency matrix (lexicographic and Gray code); they concluded that URL ordering has many advantages in spite of a small penalty in compression. In this paper we approach this issue in a more systematic way, testing some known orderings and proposing some new ones. Our experiments are made in the WebGraph framework, and show that the compression technique and the structure of the graph can produce significantly different results. In particular, we show that for a transposed web graph, URL ordering is significantly less effective, and that some new mixed orderings combining host information and Gray/lexicographic orderings outperform all previous methods: in some large transposed graphs they yield the quite incredible compression rate of 1 bit per link. We experiment with these simple ideas on some nonweb social networks and obtain results that are extremely promising and are very close to those recently achieved using shingle orderings and backlink compression schemes.

Models of Online Social Networks

Mon, 10 Oct 2011 13:58 EDT

Anthony Bonato, Noor Hadi, Paul Horn, Paweł Prałat, Changping Wang

Source: Internet Math., Volume 6, Number 3, 285--313.

Abstract:
We present a deterministic model for online social networks (OSNs) based on transitivity and local knowledge in social interactions. In the iterated local transitivity (ILT) model, at each time step and for every existing node $x$, a new node appears that joins to the closed neighbor set of $x$. The ILT model provably satisfies a number of both local and global properties that have been observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low numbers of intercommunity edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model indicating such bad expansion properties. The cop and domination numbers are shown to remain the same as those of the graph from the initial time step $G_0$, and the automorphism group of G0 is a subgroup of the automorphism group of graphs generated at all later time steps. A randomized version of the ILT model is presented that exhibits a tunable densification power-law exponent and maintains several properties of the deterministic model.

A Local Graph Partitioning Algorithm Using Heat Kernel Pagerank

Mon, 10 Oct 2011 13:58 EDT

Fan Chung

Source: Internet Math., Volume 6, Number 3, 315--330.

Abstract:
We give an improved local partitioning algorithm using heat kernel pagerank, a modified version of PageRank. For a subset $S$ with Cheeger ratio (or conductance) $h$, we show that at least a quarter of the vertices in $S$ can serve as seeds for heat kernel pagerank that lead to local cuts with Cheeger ratio at most $O(\sqrt{h})$, improving the previous bound by a factor of $\sqrt{\log s}$, where s denotes the volume of $S$.

Percolation in General Graphs

Mon, 10 Oct 2011 13:58 EDT

Fan Chung, Paul Horn, Linyuan Lu

Source: Internet Math., Volume 6, Number 3, 331--347.

Abstract:
We consider a random subgraph $G_p$ of a host graph $G$ formed by retaining each edge of $G$ with probability $p$. We address the question of determining the critical value $p$ (as a function of $G$) for which a giant component emerges. Suppose $G$ satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second-order average degree $\overline{d}$ to be $\overline{d} = \sum_v d^2_v/(\sum_v d_v)$, where $d_v$ denotes the degree of $v$. We prove that for any $\epsilon > 0$, if $p > (1 + \epsilon)/\overline{d}$, then asymptotically almost surely, the percolated subgraph $G_p$ has a giant component. In the other direction, if $p < (1 − \epsilon)/\overline{d}$, then almost surely, the percolated subgraph $G_p$ contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings. The main theorems are strengthened with much weaker assumptions.

Approximating the Number of Network Motifs

Mon, 10 Oct 2011 13:58 EDT

Mira Gonen, Yuval Shavitt

Source: Internet Math., Volume 6, Number 3, 349--372.

Abstract:
The World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species are only a few examples of systems comprising a large number of highly interconnected dynamical units. These networks contain characteristic patterns, network motifs , that occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of occurrences of network motifs as trees and bounded treewidth subgraphs of size $O(\log n)$, at most 7 for some motifs. In addition, local motif counting, counting the number of motifs in which a node participates, was recently suggested as a method of classifying nodes in the network. The premise is that the distribution of motifs in which a node participates is an indication of its function in the network. Therefore, local counting of network motifs provides a major challenge. However, no such practical algorithm exists other than local counting of triangles. We present several algorithms with time complexity $O(((3e)^k • n • |E| • \log \frac{1}{δ})/\epsilon^2)$ that approximate for every vertex the number of occurrences of the motif in which the vertex participates, for k-length cycles and $k$-length cycles with a chord, where $k = O(\log n)$, and algorithms with time complexity $O((n • |E| • \log \frac{1}{δ} )/\epsilon^2 + |E|^2 • \log n + |E| • n \log n)$ that approximate for every vertex the number of noninduced occurrences of the motif in which the vertex participates for all motifs of size four. In addition, we show algorithms that approximate the total number of occurrences of these network motifs when no efficient algorithm exists. Some of our algorithms use the “color-coding” technique.

Speeding Up Algorithms on Compressed Web Graphs

Mon, 10 Oct 2011 13:58 EDT

Chinmay Karande, Kumar Chellapilla, Reid Andersen

Source: Internet Math., Volume 6, Number 3, 373--398.

Abstract:
A variety of lossless compression schemes has been proposed to reduce the storage requirements of web graphs. One successful approach is virtual-node compression, in which often-used patterns of links are replaced by links to virtual nodes, creating a compressed graph that succinctly represents the original. In this paper, we show that several important classes of web graph algorithms can be extended to run directly on virtual-node-compressed graphs, such that their running times depend on the size of the compressed graph rather than on that of the original. These include algorithms for link analysis, estimating the size of vertex neighborhoods, and a variety of algorithms based on matrix-vector products and random walks. Similar speedups have been obtained previously for classical graph algorithms such as shortest paths and maximum bipartite matching. We measure the performance of our modified algorithms on several publicly available web graph data sets, and demonstrate significant empirical speedups that nearly match the compression ratios.

Criteria for Cluster-Based Personalized Search

Mon, 10 Oct 2011 13:58 EDT

Hyun Chul Lee, Allan Borodin

Source: Internet Math., Volume 6, Number 3, 399--435.

Abstract:
We study personalized web-ranking algorithms based on the existence of document clusterings. Motivated by topic-sensitive page ranking, we develop and implement an efficient "local-cluster" algorithm by extending the web search algorithm of Achilioptas et al., Web Search via Hub Synthesis . We propose some formal criteria for evaluating such personalized ranking algorithms and provide some preliminary experiments in support of our analysis. Both theoretically and experimentally, our algorithm differs significantly from Topic-Sensitive PageRank.

A Coupled Model for the Indegree and Outdegree Analysis of the Web

Thu, 13 Oct 2011 10:01 EDT

P. Favati, G. Lotti, O. Menchi, F. Romani

Source: Internet Math., Volume 6, Number 4, 437--459.

Abstract:
We introduce a mixed model for the Web graph that simultaneously describes the inlink and outlink distributions by taking into account the interconnection of the two processes. We derive an expression for the steady-state distribution of indegrees (outdegrees) among vertices with fixed outdegree (indegree) in terms of sums of beta functions. Experimentation on subsets of the real Web shows that the proposed distributions well reproduce the behavior of the observed data.

Nongrowing Preferential Attachment Random Graphs

Thu, 13 Oct 2011 10:01 EDT

Tomas Hruz, Ueli Peter

Source: Internet Math., Volume 6, Number 4, 461--487.

Abstract:
We consider an edge rewiring process that is widely used to model the dynamics of scale-free weblike networks. This process uses preferential attachment and operates on sparse multigraphs with n vertices and m edges. We prove that its mixing time is optimal and develop a framework that simplifies the calculation of graph properties in the steady state. The applicability of this framework is demonstrated by calculating the degree distribution, the number of self-loops, and the threshold for the appearance of the giant component.

A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees

Thu, 13 Oct 2011 10:01 EDT

Joseph Blitzstein, Persi Diaconis

Source: Internet Math., Volume 6, Number 4, 489--522.

Abstract:
Random graphs with given degrees are a natural next step in complexity beyond the Erdős–Rényi model, yet the degree constraint greatly complicates simulation and estimation. We use an extension of a combinatorial characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows for surprisingly efficient sequential importance sampling. The resulting probabilities are easily computed on the fly, allowing the user to reweight estimators appropriately, in contrast to some ad hoc approaches that generate graphs with the desired degrees but with completely unknown probabilities. Applications are given, including simulating an ecological network and estimating the number of graphs with a given degree sequence.

Euclidean versus Hyperbolic Congestion in Idealized versus Experimental Networks

Thu, 13 Oct 2011 10:02 EDT

Edmond Edmond, Mingji Lou, Francis Bonahon, Yuliy Baryshnikov

Source: Internet Math., Volume 7, Number 1, 1--27.

Abstract:
This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large $n$-dimensional hyperbolic ball $B$ of radius $R$ viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is $Θ(1)$, whereas in a Euclidean ball, the same proportion scales as $Θ(1/R^{n−1})$. This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume$^2 (B)$, whereas the same traffic load scales as volume$^{1+1/n} (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic Euclidean lattice networks. It is further conjectured that for networks that do not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of maximum traffic is near the center of mass of the network.

The Power of 1 + α for Memory-Efficient Bloom Filters

Thu, 13 Oct 2011 10:02 EDT

Evgeni Krimer, Mattan Erez

Source: Internet Math., Volume 7, Number 1, 28--44.

Abstract:
This paper presents a cache-aware Bloom-filter algorithm with improved cache behavior and lower false-positive rates compared to prior work. The algorithm relies on the power-of-two choice principle to provide a better distribution of set elements in a blocked Bloom filter. Instead of choosing a single block, we insert new elements into the less-loaded of two blocks to achieve a low false-positive rate while performing only two memory accesses on each insert or query operation. The paper also discusses an optimization technique to balance cache effectiveness with the false-positive rate to fine-tune the Bloom-filter properties.

Understanding Edge Connectivity in the Internet through Core Decomposition

Thu, 13 Oct 2011 10:02 EDT

J. Ignacio Alvarez-Hamelin, Mariano G. Beiró, Jorge R. Busch

Source: Internet Math., Volume 7, Number 1, 45--66.

Abstract:
The Internet is a complex network composed of several networks: the autonomous systems. Each of them is designed with the aim of transporting information efficiently. This information is carried over routes, which are discovered by routing protocols, such as the border gateway protocol (BGP). The protocols may find possible paths between nodes whenever they exist, or even find paths satisfying specific constraints, e.g., a certain quality of service (QoS). Here, we study connectivity as a network attribute related to both situations; we provide a formal lower bound to it based on core decomposition and low-complexity algorithms to find it. Then we apply these algorithms to analyze maps obtained from the prominent Internet mapping projects, and use the LaNet-vi open-source software for their visualization.

Graphs with Asymptotically Invariant Degree Sequences under Restriction

Thu, 13 Oct 2011 10:02 EDT

Joshua Cooper, Linyuan Lu

Source: Internet Math., Volume 7, Number 1, 67--80.

Abstract:
Scaling-free graphs are often used to describe a class of graphs that have the self-similarity property. The degree sequences of many scaling-free graphs follow the power-law distribution. In this paper,we study the distributions of graphical degree sequences that are invariant under “scaling.” We show that the invariant degree sequence must be a power-law distribution for sparse graphs if we ignore isolated vertices, or more generally, the vertices of degree less than a fixed constant k . We obtain a concentration result on the degree sequence of a random induced subgraph. The case of hypergraphs (or set systems) is also examined.

Community Structures in Classical Network Models

Thu, 13 Oct 2011 10:08 EDT

Angsheng Li, Pan Peng

Source: Internet Math., Volume 7, Number 2, 81--106.

Abstract:
Communities (or clusters) are ubiquitous in real-world networks. Researchers from different fields have proposed many definitions of communities, which are usually thought of as a subset of nodes whose vertices are well connected with other vertices in the set and have relatively fewer connections with vertices outside the set. In contrast to traditional research that focuses mainly on detecting and/or testing such clusters, we propose a new definition of community and a novel way to study community structure, with which we are able to investigate mathematical network models to test whether they exhibit the small-community phenomenon , i.e., whether every vertex in the network belongs to some small community. We examine various models and establish both positive and negative results: we show that in some models, the small-community phenomenon exists, while in some other models, it does not.

Social Influence and Evolution of Market Share

Thu, 13 Oct 2011 10:08 EDT

Simla Ceyhan, Mohammad Mousavi, Amin Saberi

Source: Internet Math., Volume 7, Number 2, 107--134.

Abstract:
We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of a number of available products. Their choice of product is a stochastic function of the inherent quality of the product and its market share. Using techniques from stochastic approximation theory, we show that market shares converge to an equilibrium. We also derive the market shares at equilibrium in terms of the level of social influence and the inherent quality of the products. In a special case, in which the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and that with strong enough social influence, monopoly occurs. These results support the observations made in the experimental study of cultural markets in Salganik et al., Experimental Study of Inequality and Unpredictability in an Artificial Cultural Market. .

Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Thu, 13 Oct 2011 10:08 EDT

Edmond Jonckheere, Poonsuk Lohsoonthorn, Fariba Ariaei

Source: Internet Math., Volume 7, Number 3, 137--177.

Abstract:
In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the thin triangle condition (TTC), to the computationally simplified, but less intuitive, four-point condition (FPC). The original motivation was that for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta measuring the thinness of a triangle scaled by its diameter must be below a certain threshold all across the graph. Here we develop various ways of scaling the 4-point delta, and develop upper bounds for the scaled 4-point delta in various spaces. A significant theoretical advantage of the TTC over the FPC is that the latter allows for a Gromov-like characterization of Ptolemaic spaces. As a major network application, it is shown that scale-free networks tend to be scaled Gromov hyperbolic, while small-world networks are rather scaled positively curved.

Equilibria and Efficiency Loss in Games on Networks

Thu, 13 Oct 2011 10:08 EDT

Joshua R. Davis, Zachary Goldman, Elizabeth N. Koch, Jacob Hilty, David Liben-Nowell, Alexa Sharp, Tom Wexler, Emma Zhou

Source: Internet Math., Volume 7, Number 3, 178--205.

Abstract:
Social networks are the substrate upon which we make and evaluate many of our daily decisions: our costs and benefits depend on whether—or how many of, or which of—our friends are willing to go to that restaurant, choose that cellular provider, already own that gaming platform. Much of the research on the “diffusion of innovation,” for example, takes a game-theoretic perspective on strategic decisions made by people embedded in a social context. Indeed, multiplayer games played on social networks, where the network's nodes correspond to the game's players, have proven to be fruitful models of many natural scenarios involving strategic interaction. In this paper, we embark on a mathematical and general exploration of the relationship between two-person strategic interactions (a “base game”) and a “networked” version of that same game. We formulate a generic mechanism for superimposing a symmetric two-player base game M on a social network G : each node of G chooses a single strategy from M and simultaneously plays that strategy against each of its neighbors in G , receiving as its payoff the sum of the payoffs from playing M against each neighbor. We denote the networked game that results by M ⊕ G . We are broadly interested in the relationship between properties of M and of M ⊕ G : how does the character of strategic interaction change when it is embedded in a social network? We focus on two particular properties: the (pure) price of anarchy and the existence of pure Nash equilibria. We show tight results on the relationship between the price of anarchy in M and M ⊕ G in coordination games. We also show that, with some exceptions when G is bipartite, the existence or absence of pure Nash equilibria (and even the guaranteed convergence of best-response dynamics) in M and M ⊕ G is not entailed in either direction. Taken together, these results suggest that the process of superimposing M on a graph is a nontrivial operation that can have rich, but bounded, effects on the strategic environment.

On the Approximability of Reachability-Preserving Network Orientations

Thu, 08 Dec 2011 13:01 EST

Michael Elberfeld, Vineet Bafna, Iftah Gamzu, Alexander Medvedovsky, Danny Segev, Dana Silverbush, Uri Zwick, Roded Sharan

Source: Internet Math., Volume 7, Number 4, 209--232.

Abstract:
We introduce a graph-orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source–target vertex pairs, the goal is to orient the graph such that a maximum number of pairs admit a directed source-to-target path. We study the complexity and approximability of this problem. We show that the problem is NP-hard even on star graphs and hard to approximate to within some constant factor. On the positive side, we provide an $Ω(log log n/ log n)$ factor approximation algorithm for the problem on n-vertex graphs. We further show that for any instance of the problem there exists an orientation of the input graph that satisfies a logarithmic fraction of all pairs and that this bound is tight up to a constant factor. Our techniques also lead to constant-factor approximation algorithms for some restricted variants of the problem.

Googling the Brain: Discovering Hierarchical and Asymmetric Network Structures, with Applications in Neuroscience

Thu, 08 Dec 2011 13:01 EST

Jonathan J. Crofts, Desmond J. Higham

Source: Internet Math., Volume 7, Number 4, 233--254.

Abstract:
Hierarchical organization is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organization. However, in many real-world networks, such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data are collated, the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall "chain of command" makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network to reveal this type of hierarchy. In doing so, we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random-graph model. This random-graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node-ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically.

Extension and Robustness of Transitivity Clustering for Protein–Protein Interaction Network Analysis

Thu, 08 Dec 2011 13:01 EST

Tobias Wittkop, Sven Rahmann, Richard Röttger, Sebastian Böcker, Jan Baumbach

Source: Internet Math., Volume 7, Number 4, 255--273.

Abstract:
Partitioning biological data objects into groups such that the objects within the groups share common traits is a longstanding challenge in computational biology. Recently, we developed and established transitivity clustering, a partitioning approach based on weighted transitive graph projection that utilizes a single similarity threshold as density parameter. In previous publications, we concentrated on the graphical user interface and on concrete biomedical application protocols. Here, we contribute the following theoretical considerations: (1) We provide proofs that the average similarity between objects from the same cluster is above the user-given threshold and that the average similarity between objects from different clusters is below the threshold. (2) We extend transitivity clustering to an overlapping clustering tool by integrating two new approaches. (3) We demonstrate the power of transitivity clustering for protein-complex detection. We evaluate our approaches against others by utilizing gold-standard data that was previously used by Brohée et al. for reviewing existing bioinformatics clustering tools. The extended version of this article is available online at http://transclust.mpi-inf. mpg.de.

Using Biological Networks in Protein Function Prediction and Gene Expression Analysis

Thu, 08 Dec 2011 13:01 EST

Limsoon Wong

Source: Internet Math., Volume 7, Number 4, 274--298.

Abstract:
While sequence homology search has been the main workhorse in protein function prediction, it is not applicable to a significant portion of novel proteins that do not have informative homologues in sequence databases. Similarly, while statistical tests and learning algorithms based purely on gene expression profiles have been popular for analyzing disease samples, critical issues remain in the understanding of diseases based on the differentially expressed genes suggested by these methods. In the past decade, a large number of databases providing information on various types of biological networks have become available. These databases make it possible to tackle these and other biological problems in novel ways. This paper presents a review of biological network databases and approaches to protein function prediction and gene expression profile analysis that are based on biological networks.

KeyPathwayMiner: Detecting Case-Specific Biological Pathways Using Expression Data

Thu, 08 Dec 2011 13:01 EST

Nicolas Alcaraz, Hande Kücük, Jochen Weile, Anil Wipat, Jan Baumbach

Source: Internet Math., Volume 7, Number 4, 299--313.

Abstract:
Recent advances in systems biology have provided us with massive amounts of pathway data that describe the interplay of genes and their products. The resulting biological networks can be modeled as graphs. By means of "omics" technologies, such as microarrays, the activity of genes and proteins can be measured. Here, data from microarray experiments is integrated with the network data to gain deeper insights into gene expression. We introduce KeyPathwayMiner, a method that enables the extraction and visualization of interesting subpathways given the results of a series of gene expression studies. We aim to detect highly connected subnetworks in which most genes or proteins show similar patterns of expression. Specifically, given network and gene expression data, KeyPathwayMiner identifies those maximal subgraphs where all but k nodes of the subnetwork are expressed similarly in all but l cases in the gene expression data. Since identifying these subgraphs is computationally intensive, we developed a heuristic algorithm based on Ant Colony Optimization. We implemented KeyPathwayMiner as a plug-in for Cytoscape. Our computational model is related to a strategy presented by Ulitsky et al. in 2008. Consequently, we used the same data sets for evaluation. KeyPathwayMiner is available online at http://keypathwayminer.mpi-inf.mpg.de.

NAViGaTOR: Large Scalable and Interactive Navigation and Analysis of Large Graphs

Thu, 08 Dec 2011 13:01 EST

Amira Djebbari, Muhammad Ali, David Otasek, Max Kotlyar, Kristen Fortney, Serene Wong, Anthony Hrvojic, Igor Jurisica

Source: Internet Math., Volume 7, Number 4, 314--347.

Abstract:
Network visualization tools offer features enabling a variety of analyses to satisfy diverse requirements. Considering complexity and diversity of data and tasks, there is no single best layout, no single best file format or visualization tool: one size does not fit all. One way to cope with these dynamics is to support multiple scenarios and workflows. NAViGaTOR (Network Analysis, Visualization & Graphing TORonto) offers a complete system to manage diverse workflows from one application. It allows users to manipulate large graphs interactively using an innovative graphical user interface (GUI) and through fast layout algorithms with a small memory footprint. NAViGaTOR facilitates integrative network analysis by supporting not only visualization but also visual data mining.

Geometric Protean Graphs

Thu, 31 May 2012 20:58 EDT

Anthony Bonato, Jeannette Janssen, Paweł Prałat

Source: Internet Math., Volume 8, Number 1, 2--28.

Abstract:
We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs, nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power-law degree distributions, the small-world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.

Constant Price of Anarchy in Network-Creation Games via Public-Service Advertising

Thu, 31 May 2012 20:58 EDT

Erik D. Demaine, Morteza Zadimoghaddam

Source: Internet Math., Volume 8, Number 1, 29--45.

Abstract:
Network-creation games have been studied recently in many different settings. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without paying much to construct the network. Many generalizations have been considered, including nonuniform interests between nodes, general graphs of allowable edges, and bounded-budget agents. In all of these settings, there is no known constant bound on the price of anarchy. In fact, in many cases, the price of anarchy can be very large, namely, a constant power of the number of agents. This means that we have no control over the behavior of a network when agents act selfishly. On the other hand, the price of stability in all these models is constant, which means that there is chance that agents act selfishly and we end up with a reasonable social cost. In this paper, we show how to use an advertising campaign (as introduced in [Balcan et al., “Improved Equilibria via Public Service Advertising.”) to find efficient equilibria in an $(n, k)$- uniform bounded-budget connection game; our result holds for $k = \omega(\log(n))$. More formally, we present advertising strategies such that if an $\alpha$ fraction of the agents agree to cooperate in the campaign, the social cost will be at most $O(1/\alpha)$ times the optimum cost. This is the first constant bound on the price of anarchy that interestingly can be adapted to different settings. We also generalize our method to work in cases in which $\alpha$ is not known in advance. Also, we do not need to assume that the cooperating agents spend all their budget on the campaign; even a small fraction ($\beta$ fraction) of their budget is sufficient to obtain a constant price of anarchy.

Finding and Visualizing Graph Clusters Using PageRank Optimization

Thu, 31 May 2012 20:58 EDT

Fan Chung, Alexander Tsiatas

Source: Internet Math., Volume 8, Number 1, 46--72.

Abstract:
We give algorithms for finding graph clusters and drawing graphs, highlighting local community structure within the context of a larger network. For a given graph G, we use the personalized PageRank vectors to determine a set of clusters, by optimizing the jumping parameter α subject to several cluster variance measures in order to capture the graph structure according to PageRank. We then give a graph visualization algorithm for the clusters using PageRank-based coordinates. Several drawings of real-world data are given, illustrating the partition and local community structure.

Fast Matrix Computations for Pairwise and Columnwise Commute Times and Katz Scores

Thu, 31 May 2012 20:58 EDT

Francesco Bonchi, Pooya Esfandiar, David F. Gleich, Chen Greif, Laks V. S. Lakshmanan

Source: Internet Math., Volume 8, Number 1, 73--112.

Abstract:
We explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pairwise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise commute-time method and columnwise Katz algorithm both have attractive theoretical properties and empirical performance.

Multiplicative Attribute Graph Model of Real-World Networks

Thu, 31 May 2012 20:58 EDT

Myunghwan Kim, Jure Leskovec

Source: Internet Math., Volume 8, Number 1, 113--160.

Abstract:
Networks are a powerful way to describe and represent social, technological, and biological systems, where nodes represent entities (people, web sites, genes) and edges represent interactions (friendships, communication, regulation). The study of such networks then seeks to find common structural patterns and explain their emergence through tractable models of network formation. In most networks, each node is associated with a rich set of attributes or features. For example, users in online social networks have profile information, genes have properties and functions, and web pages contain text. However, most existing network models focus on modeling the network structure while ignoring the features and properties of the nodes. Thus, the questions that we address in this work are as follows: What is a mathematically tractable model that naturally captures ways in which the network structure and node attributes interact? What are the properties of networks that arise under such a model? We present a model of network structure that we refer to as the multiplicative attribute graphs (MAG) model. The MAG model naturally captures the interactions between the network structure and the node attributes. We consider a model in which each node has a vector of categorical attributes associated with it. The link-affinity matrix then models the interaction between the value of a particular attribute and the probability of a link between a pair of nodes. The MAG model yields itself to mathematical analysis, and we derive thresholds for the connectivity and the emergence of the giant connected component, and show that the model gives rise to networks with a constant diameter. We also analyze the degree distribution and find surprising flexibility of the MAG model in that it can generate networks with either log-normal or power-law degree distribution.

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Thu, 31 May 2012 20:58 EDT

Mihail N. Kolountzakis, Gary L. Miller, Richard Peng, Charalampos E. Tsourakakis

Source: Internet Math., Volume 8, Number 1, 161--185.

Abstract:
The number of triangles is a computationally expensive graph statistic frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model), and in important real-world applications such as spam detection, uncovering the hidden thematic structures in the Web, and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms that run fast, use little space, provide accurate estimates of the number of triangles, and preferably are parallelizable. In this paper we present an efficient triangle-counting approximation algorithm that can be adapted to the semistreaming model. Its key idea is to combine the sampling algorithm of Tsourakakis, and the partitioning of the set of vertices into high- and low-degree subsets as in Alon, treating each set appropriately. From a mathematical perspective, we present a simplified proof of Tsourakakis, “Counting Triangles Using Projections,” that uses the powerful Kim–Vu concentration inequality based on the Hajnal–Szemerédi theorem. Furthermore, we improve bounds of existing triple-sampling techniques based on a theorem of Ahlswede and Katona. We obtain a running time $O(m + \frac{m^{3 / 2} \log n} {t\epsilon^2})$ and an $(1 ±\epsilon)$ approximation, where $n$ is the number of vertices, $m$ is the number of edges, and $\Delta$ is the maximum number of triangles in which any single edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage $O(m^{1/2} \log n + \frac{m^{3 / 2} \log n} {t\epsilon^2} )$ and a constant number of passes (three) over the graph stream. We apply our methods to various networks with several millions of edges and we obtain excellent results, outperforming existing triangle-counting methods. Finally, we propose a random-projection-based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

Geometric Protean Graphs

Thu, 14 Jun 2012 08:49 EDT

Anthony Bonato, Jeannette Janssen, Paweł Prałat

Source: Internet Math., Volume 8, Number 1-2, 2--28.

Constant Price of Anarchy in Network-Creation Games via Public-Service Advertising

Thu, 14 Jun 2012 08:49 EDT

Erik D. Demaine, Morteza Zadimoghaddam

Source: Internet Math., Volume 8, Number 1-2, 29--45.

Finding and Visualizing Graph Clusters Using PageRank Optimization

Thu, 14 Jun 2012 08:49 EDT

Fan Chung, Alexander Tsiatas

Source: Internet Math., Volume 8, Number 1-2, 46--72.

Fast Matrix Computations for Pairwise and Columnwise Commute Times and Katz Scores

Thu, 14 Jun 2012 08:49 EDT

Francesco Bonchi, Pooya Esfandiar, David F. Gleich, Chen Greif, Laks V. S. Lakshmanan

Source: Internet Math., Volume 8, Number 1-2, 73--112.

Multiplicative Attribute Graph Model of Real-World Networks

Thu, 14 Jun 2012 08:49 EDT

Myunghwan Kim, Jure Leskovec

Source: Internet Math., Volume 8, Number 1-2, 113--160.

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Thu, 14 Jun 2012 08:49 EDT

Mihail N. Kolountzakis, Gary L. Miller, Richard Peng, Charalampos E. Tsourakakis

Source: Internet Math., Volume 8, Number 1-2, 161--185.

Monotone Graph Limits and Quasimonotone Graphs

Tue, 21 Aug 2012 16:30 EDT

Bála Bollobás, Svante Janson, Oliver Riordan

Source: Internet Math., Volume 8, Number 3, 187--231.

Abstract:
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object that may be represented by a symmetric function $W$ on $[0, 1]^2$ , i.e., a kernel or graphon . In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a “quasimonotonicity” property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1 −W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.

Moment-Based Estimation of Stochastic Kronecker Graph Parameters

Tue, 21 Aug 2012 16:30 EDT

David F. Gleich, Art B. Owen

Source: Internet Math., Volume 8, Number 3, 232--256.

Abstract:
Stochastic Kronecker graphs supply a parsimonious model for large sparse real-world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have, however, proved difficult to choose in specific applications. This article looks at method-of-moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast, and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement is especially prominent for the number of triangles in the graph.

Degree Distribution and Number of Edges between Nodes of Given Degrees in the Buckley–Osthus Model of a Random Web Graph

Tue, 21 Aug 2012 16:30 EDT

Evgeniy A. Grechnikov

Source: Internet Math., Volume 8, Number 3, 257--287.

Abstract:
In this paper, we study some important statistics of the random graph $H^{(t)}_{a ,k}$ in the Buckley-Osthus model, where $t$ is the number of nodes, $kt$ is the number of edges (so that $k \in \mathbb{N}$), and $a \gt 0$ is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás-Riordan model. First, we find a new asymptotic formula for the expectation of the number $R(d, t)$ of nodes of a given degree $d$ in a graph in this model. Such a formula is known for $a \in \mathbb{N}$ and $d \le t^{1/100(a+1)}$ . Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities $R(d_1 , t)$ and $R(d_2 , t)$, and using the second-moment method we show that $R(d, t)$ is tightly concentrated around its mean for all possible values of $d$ and $t$. Furthermore, we study a more complicated statistic of the web graph: $X(d_1, d_2 , t)$ is the total number of edges between nodes whose degrees are equal to $d_1$ and $d_2$ respectively. We also find an asymptotic formula for the expectation of $X(d_1, d_2 , t)$ and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of $d_1 , d_2$ , and $t$.

An Extended Stochastic Model for Quantitative Security Analysis of Networked Systems

Tue, 21 Aug 2012 16:30 EDT

Maochao Xu, Shouhuai Xu

Source: Internet Math., Volume 8, Number 3, 288--320.

Abstract:
Quantitative security analysis of networked computer systems has been an open problem in computer security for decades. Recently, a promising approach was proposed in Li et al., which, however, made some strong assumptions including the exponential distribution of, and the independence among, the relevant random variables. In this paper, we substantially weaken these assumptions while offering, in addition to the same types of analytical results as in Li et al., methods for obtaining the desired security quantities in practice. Moreover, we investigate the problem from a higher-level abstraction, which also leads to both analytical results and practical methods for obtaining the desired security quantities. These should represent a significant step toward ultimately solving the problem of quantitative security analysis of networked computer systems.

Mean Commute Time for Random Walks on Hierarchical Scale-Free Networks

Thu, 06 Dec 2012 11:06 EST

Yilun Shang

Source: Internet Math., Volume 8, Number 4, 321--337.

Abstract:
In recent years, there has been a surge of research interest in networks with scale-free topologies, partly due to the fact that they are prevalent in scientific research and real-life applications. In this paper, we study random-walk issues on a family of two-parameter scale-free networks, called $(x, y)$-flowers. These networks, which are constructed in a deterministic recursive fashion, display rich behaviors such as the small-world phenomenon and pseudofractal properties. We derive analytically the mean commute times for random walks on $(x, y)$-flowers and show that the mean commute times scale with the network size as a power-law function with exponent governed by both parameters $x$ and $y$. We also determine the mean effective resistance and demonstrate that it changes sharply between different choices of $x$ and $y$. Furthermore, we compare mean commute times for $(x, y)$-flowers with those for Erdős–Rényi random graphs. Our theoretical results are verified by numerical studies.

Model Selection for Social Networks Using Graphlets

Thu, 06 Dec 2012 11:06 EST

Jeannette Janssen, Matt Hurshman, Nauzer Kalyaniwalla

Source: Internet Math., Volume 8, Number 4, 338--363.

Abstract:
Several network models have been proposed to explain the link structure observed in online social networks. This paper addresses the problem of choosing the model that best fits a given real-world network. We implement a model-selection method based on unsupervised learning. An alternating decision tree is trained using synthetic graphs generated according to each of the models under consideration. We use a broad array of features, with the aim of representing different structural aspects of the network. Features include the frequency counts of small subgraphs (graphlets) as well as features capturing the degree distribution and small-world property. Our method correctly classifies synthetic graphs, and is robust under perturbations of the graphs. We show that the graphlet counts alone are sufficient in separating the training data, indicating that graphlet counts are a good way of capturing network structure. We tested our approach on four Facebook graphs from various American universities. The models that best fit these data are those that are based on the principle of preferential attachment.

Balance in Random Signed Graphs

Thu, 06 Dec 2012 11:06 EST

A. El Maftouhi, Y. Manoussakis, O. Megalakaki

Source: Internet Math., Volume 8, Number 4, 364--380.

Abstract:
By extending Heider's and Cartwright–Harary's theory of balance in deterministic social structures, we study the problem of balance in social structures in which relations among individuals are random. An appropriate model for representing such structures is that of random signed graphs $G_{n,p,q}$, defined as follows. Given a set of $n$ vertices and fixed numbers $p$ and $q$, $0 \lt p + q \lt 1$, then between each pair of vertices, there exists a positive edge, a negative edge, or no edge with respective probabilities $p, q, 1 − p − q$. We first show that almost always (i.e., with probability tending to 1 as $n \to \infty$), the random signed graph $G_{n,p,q}$ is unbalanced. Subsequently we estimate the maximum order of a balanced induced subgraph in $G_{n,p,p}$ and show that its order achieves only a finite number of values. Next, we study the asymptotic behavior of the degree of balance and give upper and lower bounds for the line index of balance. Finally, we study the threshold function of balance, e.g., a function $p_0 (n)$ such that if $p \gg p_0 (n)$, then the random signed graph $G_{n,p,p}$ is almost always unbalanced, and otherwise, it is almost always balanced.

Digraph Laplacian and the Degree of Asymmetry

Thu, 06 Dec 2012 11:06 EST

Yanhua Li, Zhi-Li Zhang

Source: Internet Math., Volume 8, Number 4, 381--401.

Abstract:
In this paper we extend and generalize the standard spectral graph theory (or random-walk theory) on undirected graphs to digraphs. In particular, we introduce and define a normalized digraph Laplacian ( Diplacian for short) $\Gamma$ for digraphs, and prove that (1) its Moore–Penrose pseudoinverse is the discrete Green’s function of the Diplacian matrix as an operator on digraphs, and (2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive a new formula for computing hitting and commute times in terms of the Moore–Penrose pseudoinverse of the Diplacian, or equivalently, the singular values and vectors of the Diplacian. Furthermore, we show that the Cheeger constant defined in “Laplacians and the Cheeger Inequality for Directed Graphs,” is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric, the largest singular value of the skewed Laplacian $\nabla = (\Gamma − \Gamma^T )/2$, to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound than that in “Laplacians and the Cheeger Inequality for Directed Graphs,” on the Markov chain mixing rate, and a bound on the second-smallest singular value of $\Gamma$.

Bistability through Triadic Closure

Thu, 06 Dec 2012 11:06 EST

Peter Grindrod, Desmond J. Higham, Mark C. Parsons

Source: Internet Math., Volume 8, Number 4, 402--423.

Abstract:
We propose and analyze a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, online social behavior, and information processing in neuroscience. We model the evolving network as a discrete-time Markov chain and study a very general framework in which edges conditioned on the current state appear or disappear independently at the next time step. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cellphone data, and find support for a block model in which individuals are assigned to one of two distinct groups having different within-group and across-group dynamics.

Strategic Coloring of a Graph

Thu, 06 Dec 2012 11:06 EST

Bruno Escoffier, Laurent Gourvès, Jérôme Monnot

Source: Internet Math., Volume 8, Number 4, 424--455.

Abstract:
We study a strategic game in which every node of a graph is owned by a player who has to choose a color. A player’s payoff is 0 if at least one neighbor selected the same color; otherwise, it is the number of players who selected the same color. The social cost of a state is defined as the number of distinct colors that the players use. It is ideally equal to the chromatic number of the graph, but it can substantially deviate because every player cares about his own payoff, however bad the social cost may be. Following previous work in Panagopoulou and Spirakis, “A Game Theoretic Approach for Efficient Graph Coloring,” on the Nash equilibria of the coloring game, we give worst-case bounds on the social cost of stable states. Our main contribution is an improved (tight) bound for the worst-case social cost of a Nash equilibrium, as well as the study of strong equilibria, their existence, and how far they are from social optima.

Approximations of the Generalized Inverse of the Graph Laplacian Matrix

Thu, 06 Dec 2012 11:06 EST

Enrico Bozzo, Massimo Franceschet

Source: Internet Math., Volume 8, Number 4, 456--481.

Abstract:
We devise methods for finding approximations of the generalized inverse of the graph Laplacian matrix, which arises in many graph-theoretic applications. Finding this matrix in its entirety involves solving a matrix inversion problem, which is resourcedemanding in terms of consumed time and memory and hence impractical whenever the graph is relatively large. Our approximations use only a few eigenpairs of the Laplacian matrix and are parametric with respect to this number, so that the user can compromise between effectiveness and efficiency of the approximate solution. We apply the devised approximations to the problem of computing current-flow betweenness centrality on a graph. However, given the generality of the Laplacian matrix, many other applications can be sought. We experimentally demonstrate that the approximations are effective already with a constant number of eigenpairs. These few eigenpairs can be stored with a linear amount of memory in the number of nodes of the graph, and in the realistic case of sparse networks, they can be efficiently computed using one of the many methods for retrieving a few eigenpairs of sparse matrices that abound in the literature.