# Advances in Network Analysis and its Applications by Hamed Amini, Andreea Minca (auth.), Evangelos Kranakis

By Hamed Amini, Andreea Minca (auth.), Evangelos Kranakis (eds.)

As good as highlighting probably worthwhile purposes for community research, this quantity identifies new objectives for mathematical study that promise to supply insights into community structures concept in addition to facilitating the cross-fertilization of principles among sectors. targeting monetary, defense and social facets of networking, the amount provides to the transforming into physique of facts displaying that community research has functions to transportation, communique, overall healthiness, finance, and social coverage extra generally. It offers strong types for knowing the habit of advanced platforms that, in flip, will impression a variety of state-of-the-art sectors in technological know-how and engineering, akin to instant conversation, community safeguard, dispensed computing and social networking, monetary research, and cyber warfare.

The quantity bargains an insider’s view of state-of-the-art study in community structures, together with methodologies with sizeable power for interdisciplinary software. The individuals have all awarded fabric at a sequence of workshops prepared on behalf of Canada’s MITACS initiative, which cash initiatives and learn offers in ‘mathematics for info expertise and intricate systems’. those lawsuits comprise papers from workshops on monetary networks, community defense and cryptography, and social networks. MITACS has proven that the partially ghettoized nature of community platforms learn has ended in duplicated paintings in discrete fields, and hence this initiative has the capability to avoid wasting time and speed up the velocity of analysis in a few components of community structures research.

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**Example text**

Then for all i = 1, . . s. #{ j ∈ v, Wn (i, j) > 0} = dn− (i). We denote by αn (Wn , qn ) the set of defaults at the end of the cascade generated by the set of nodes {i | qn (i) = 0}. The following theorems give the asymptotic behavior of this quantity. 5. 18) where Bin( j, π) denotes a binomial variable with parameters j and π. 5. , π∗ = inf{π ∈ [0, 1] | I(π) = π}. 1. , if I(π) > π for all π ∈ [0, 1), then asymptotically all nodes switch to state 1: p αn (Wn , qn ) → 1. 2. If π∗ < 1 and furthermore π∗ is a stable fixed point of I (I (π∗ ) < 1), then the asymptotic fraction of nodes in state 1 at the end of the cascade satisfies: 1 Mathematical Modeling of Systemic Risk 23 j αn (Wn , qn ) → ∑ µ( j, k) ∑ p( j, k, θ)P(Bin( j, π∗ ) ≥ θ).

The mechanism of shock transmission is treated differently by GK and by NYYA, and this is an important distinction between the models. 1 Shock Transmission in the GK Model In the GK model, defaulted banks do not repay any portion of their outstanding interbank debts because the timescale for any recovery on these defaulted loans is assumed to exceed the timescale of the contagion spread in the system. Consequently, all creditors of a bank which defaulted in timestep n receive, at timestep n + 1, a shock of magnitude equal to the total size of their loan to the defaulted bank.

1). This process of shock transmission continues until there are no new defaults, at which point the cascade terminates. 2 Shock Transmission in the NYYA Model The NYYA model allows for the possibility of non-zero recovery on defaulted loans. , σ > c jk . 1). Thus the shock transmitted to each creditor of the defaulted bank is min σ − c jk ,w . P. Gleeson et al. As in the GK model, shocks transmitted from banks which default at timestep n will affect the creditor banks at timestep n + 1, and a cascade of banks failures may ensue.