We are concerned with an appropriate mathematical measure of resilience in the face of targeted node attacks for arbitrary degree networks, and subsequently comparing the resilience of different scale-free network models with the proposed measure. We strongly motivate our resilience measure termed (VAT), which is denoted mathematically as @math , where @math is the largest connected component in @math . We attempt a thorough comparison of VAT with several existing resilience notions: conductance, vertex expansion, integrity, toughness, tenacity and scattering number. Our comparisons indicate that for artbitrary degree distributions VAT is the only measure that fully captures both the major of a network and the resulting upon targeted node attacks (both captured in a manner proportional to the size of the attack set). For the case of @math -regular graphs, we prove that @math , where @math is the conductance of the graph @math . Conductance and expansion are well-studied measures of robustness and bottlenecks in the case of regular graphs but fail to capture resilience in the case of highly heterogeneous degree graphs. Regarding comparison of different scale-free graph models, our experimental results indicate that PLOD graphs with degree distributions identical to BA graphs of the same size exhibit consistently better vertex attack tolerance than the BA type graphs, although both graph types appear asymptotically resilient for BA generative parameter @math . BA graphs with @math also appear to lack resilience, not only exhibiting very low VAT values, but also great transparency in the identification of the vulnerable node sets, namely the hubs, consistent with well known previous work.
It is important to note, of course, that conductance (in its various normalized and non-normalized forms) is a fundamental property of graphs independent of its applicability to edge-based resilience or resilience in general. Conductance is intimately related to both the mixing times of random walks on graphs @cite_21 and the eigenvalues of the graph's adjacency matrix (or normalized adjacency matrix as befits the situation) @cite_4 . Therefore, one very important application of conductance has involved the analysis of randomized algorithms, particularly for hard counting problems, yielding an important place for the measure in theoretical computer science. Many of the Markov chains considered in such contexts are regular or almost-regular, and it is relatively recently that the heterogeneity of many actual networks has become established across many fields, the relevance of such degree distributions first being established not by theoretical computer scientists but by in any case.
@cite_21 @cite_4
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In local community detection by seed expansion a single cluster concentrated around few given query nodes in a graph is discovered in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. Algorithms that directly optimize conductance usually add or remove one node at a time to find a local optimum. This amounts to fix a specific neighborhood structure over clusters. A natural way to avoid the problem of choosing a specific neighborhood structure is to use a continuous relaxation of conductance. This paper studies such a continuous relaxation of conductance. We show that in this setting continuous optimization leads to hard clusters. We investigate the relation of conductance with weighted kernel k-means for a single cluster, which leads to the introduction of a weighted kernel 1-mean objective function, called -conductance, where is a parameter which influences the size of the community. Conductance is obtained by setting to 0. Two algorithms for local optimization of -conductance based on the expectation maximization and the projected gradient descend method are developed, called EMc and PGDc, respectively. We show that for =0 EMc corresponds to gradient descend with an infinite step size at each iteration. We design a procedure to automatically select a value for . Performance guarantee for these algorithms is proven for a class of dense communities centered around the seeds and well separated from the rest of the network. On this class we also prove that our algorithms stay localized. A comparative experimental analysis on networks with ground-truth communities is performed using state-of-the-art algorithms based on the graph diffusion method. Our experiments indicate that EMc and PGDc stay localized and produce communities most similar to the ground. One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon. Let G=(V,E), be an undirected, connected n vertex graph with m edges. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex of an n vertex graph. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our main result is that C(n)= O(1 (1-lambda_2))*((log n)^4 +n A)), where lambda_2 is the absolute value of the second largest eigenvalue of the transition matrix of the random walk, A= (sum d^2(v)) (d^2 n), d(v) is the degree of vertex v, and d is the average node degree. The parameter A is an indicator of the variability of node degrees. Thus 1 <= A =O(n), with A=1 for regular graphs. We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation |p n (x,y) π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.